Question: You have found the following ages (in years) of all 4 zebras at your local zoo: $ 24,\enspace 3,\enspace 16,\enspace 7$ What is the average age of the zebras at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we have data for all 4 zebras at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{24 + 3 + 16 + 7}{{4}} = {12.5\text{ years old}} $ Find the squared deviations from the mean for each zebra. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $24$ years $11.5$ years $132.25$ years $^2$ $3$ years $-9.5$ years $90.25$ years $^2$ $16$ years $3.5$ years $12.25$ years $^2$ $7$ years $-5.5$ years $30.25$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{132.25} + {90.25} + {12.25} + {30.25}} {{4}} $ $ {\sigma^2} = \dfrac{{265}}{{4}} = {66.25\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{66.25\text{ years}^2}} = {8.1\text{ years}} $ The average zebra at the zoo is 12.5 years old. There is a standard deviation of 8.1 years.